Module 6: Free Convections Lecture 25: Buoyancy Driven Flows. The Lecture Contains: Free Convection. Objectives_template
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1 The Lecture Contains: Free Convection file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_1.html[12/24/2014 6:07:31 PM]
2 Introduction Now we shall consider situations for which there is no forced velocity, yet convection currents exist within the fluid. They originate when a body force acts on a fluid in which there are density gradients. The net effect is a buoyancy force, which induces fluid velocity. The density gradient is due to a temperature gradient, and the body force is due to the gravitational field. The book by Gebhart, Jaluria et al.(1988) provides excellent insight in to natural convective flows and its importance. Figure 6.1: Boundary layer development on a heated vertical plate file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_2.html[12/24/2014 6:07:31 PM]
3 Free convection over a vertical flat plate We focus on a classical example of boundary layer development on a heated vertical plate. The fluid close to plate is less dense than fluid that is further removed. Buoyancy forces therefore induce a free convection boundary layer in which the heated fluid rises vertically,entraining fluid from quiescent region. The resulting velocity distribution is unlike that associated with forced convection boundary layers. In particular the velocity is zero as as well as at outside the boundary layer, we can write substituting (6.2) into (6.1), we obtain the following equation. Introducing the volume coefficient of expansion volume,we can write file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_3.html[12/24/2014 6:07:31 PM]
4 From (6.3) and (6.4) we can write (6.5) Like forced flow boundary layer, y momentum equation is The set of governing equation is now (6.6) (6.7) (6.8) with [this might be needed for calculating] (for perfect gas) Boundary condition file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_4.html[12/24/2014 6:07:31 PM]
5 Integration (6.7) within the boundary layer yields or (6.9) integration of equations (6.8) yields or, file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_5.html[12/24/2014 6:07:31 PM]
6 (6.10) Defining and substituting this in equation (6.10), yields (6.11) The temperature distribution may be assumed as (6.12) This satisfies file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_6.html[12/24/2014 6:07:31 PM]
7 The velocity in the boundry layer is given by (6.13) where is some arbitrary function of x. The boundary conditions are These boundary conditions will finally produce Substituting these values in the polynomial we get (6.14) file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_7.html[12/24/2014 6:07:32 PM]
8 The terms can be included in so that we can write (6.15) The maximum velocity and its position (distance in y direction) at any x can be obtained from equation (6.15) (6.16) (6.17) but velocity is zero Therefore, the velocity is maximum or, file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_8.html[12/24/2014 6:07:32 PM]
9 Energy equation (6.11) can be now written as (6.18) The momentum equation (6.9) may be written as or, (6.19) In equation (6.18) and (6.19), and are dependent variables and x is the independent variable. To solve the above two equations, and are considered as function of x file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_9.html[12/24/2014 6:07:32 PM]
10 Assume and very according to the following functions :- Substitution in (6.18) will produce (6.20) Substitution in (6.19) will produce (6.21) The equations have to be treated in the following way. First let us consider equation (6.20). If a monomial nonlinear expression is dependent on a variable (say x) that is equivalent to an expression independent of x, then for universal validity of the equation as a function of x,the power of the nonlinear expression must vanish.if we divide entire equation (6.20) by its right hand side, then for the universal validity of the expression, we can say file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_10.html[12/24/2014 6:07:32 PM]
11 Now, let us look at equation (6.21). Any number of non-zero degree can combine to result into one term, provided each of the terms has same degree. The degree of the resulting term will also be equal to each of the terms. This will lead to The conclusion derived from equation (6.20) and one of the two-conclusions derived from equation (6.21) is same. Now by solving these, we get Substituting the values of m and n in (6.20) and (6.21), we get (6.22) (6.23) Solving the above two equations, we get (6.24) file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_11.html[12/24/2014 6:07:32 PM]
12 (6.25) Substituting for evaluating and and substituting for Introducing Grashof number we get (6.26) Equation (6.26) gives the variation of boundary layer thickness along the height of the plate. Here thermal boundary layer thickness and hydrodynamic boundary layer thickness are same. file:///d /Web%20Course%20(Ganesh%20Rana)/Dr.%20gautam%20biswas/Final/convective_heat_and_mass_transfer/lecture25/25_12.html[12/24/2014 6:07:32 PM]
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